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In the figure above, ADC is an equilateral triangle. AC is the angle b
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20 Jun 2023, 00:50
20 Jun 2023, 00:50
Expert Reply
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90% (01:27) correct
10% (01:54) wrong based on 10 sessions
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GRe In the figure above, ADC is an equilateral triangle. AC.jpg [ 10.07 KiB | Viewed 1208 times ]
In the figure above, ADC is an equilateral triangle. AC is the angle bisector of ∠BCD. If BC = 2, what is the perimeter of quadrilateral ABCD?
A. \(6+\sqrt{3}\)
B. \(8+\sqrt{3}\)
C. \(10+\sqrt{3}\)
D. \(6+2 \sqrt{3}\)
E. \(10+2 \sqrt{3}\)
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Re: In the figure above, ADC is an equilateral triangle. AC is the angle b
[#permalink]
16 Jul 2023, 10:46
16 Jul 2023, 10:46
Expert Reply
OE
Given that, ∆ ADC is an equilateral triangle.
So, ∠ ACD = 60°
It is given that; AC is the angle bisector of the ∠ BCD.
That implies, ∠ BCA = ∠ ACD.
Hence,
∠ BCA = 60°
Now, in ⊿ ABC,
∠ ABC = 90°
, ∠ BCA = 600°
so ∠ BAC = 30° (Sum of all angles in a triangle is 180°)
Hence, ⊿ ABC is a 30 – 60 – 90 triangle.
We know that the sides opposite to angles 30o
- 60°
- 90° are in the ratio 1: √3 : 2
respectively.
that is, BC: AB: AC
1: √3: 2
2: 2√3: 4 (Multiplying every term by 2)
Hence, if BC = 2 then AB = 2√3 and AC = 4.
As ⊿ ACD is an equilateral triangle, AC = AD = CD = 4.
Hence, Perimeter of quadrilateral ABCD = AB + BC + CD + AD
1. = 2√3 + 2 + 4 + 4
2. = 10 + 2√3
E is the answer
Given that, ∆ ADC is an equilateral triangle.
So, ∠ ACD = 60°
It is given that; AC is the angle bisector of the ∠ BCD.
That implies, ∠ BCA = ∠ ACD.
Hence,
∠ BCA = 60°
Now, in ⊿ ABC,
∠ ABC = 90°
, ∠ BCA = 600°
so ∠ BAC = 30° (Sum of all angles in a triangle is 180°)
Hence, ⊿ ABC is a 30 – 60 – 90 triangle.
We know that the sides opposite to angles 30o
- 60°
- 90° are in the ratio 1: √3 : 2
respectively.
that is, BC: AB: AC
1: √3: 2
2: 2√3: 4 (Multiplying every term by 2)
Hence, if BC = 2 then AB = 2√3 and AC = 4.
As ⊿ ACD is an equilateral triangle, AC = AD = CD = 4.
Hence, Perimeter of quadrilateral ABCD = AB + BC + CD + AD
1. = 2√3 + 2 + 4 + 4
2. = 10 + 2√3
E is the answer
gmatclubot
Re: In the figure above, ADC is an equilateral triangle. AC is the angle b [#permalink]
16 Jul 2023, 10:46
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